Partial Reductions for Kleene Algebra with Linear Hypotheses

2026-01-20

AI Summary

In Kleene algebra with linear hypotheses, conventional approaches struggle to automatically construct reduction maps satisfying regularity constraints, thereby limiting the scope of provable program equivalences. This work proposes a formal automata-based method that introduces the notion of partial reductions, achieving local completeness within the reduction domain and circumventing the stringent regularity requirements imposed by total reductions. By integrating automata theory with formal language techniques, the approach establishes a mechanized framework for generating reductions, enabling the automatic derivation of a broader class of program equivalences. This significantly enhances both the decidability and practical applicability of existing verification systems.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Kleene algebra (KA) is an important tool for reasoning about general program equivalences, with a decidable and complete equational theory. However, KA cannot always prove equivalences between specific programs. For this purpose, one adds hypotheses to KA that encode program-specific knowledge. Traditionally, a map on regular expressions called a reduction then lets us lift decidability and completeness to these more expressive systems. Explicitly constructing such a reduction requires significant labour. Moreover, due to regularity constraints, a reduction may not exist for all combinations of expression and hypothesis. We describe an automaton-based construction to mechanically derive reductions for a wide class of hypotheses. These reductions can be partial, in which case they yield partial completeness: completeness for expressions in their domain. This allows us to automatically establish the provability of more equivalences than what is covered in existing work.

Problem

• Kleene algebra
• linear hypotheses
• partial reductions
• program equivalences
• regular expressions

Innovation

• automaton-based construction
• partial reductions
• Kleene algebra
• linear hypotheses
• partial completeness